Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Galton-Watson theta-processes in a varying environment

Serik Sagitov, Yerakhmet Zhumayev

Abstract

We consider a special class of Galton-Watson theta-processes in a varying environment fully defined by four parameters, with two of them $(θ,r)$ being fixed over time $n$, and the other two $(a_n,c_n)$ characterizing the altering reproduction laws. We establish a sequence of transparent limit theorems for the theta-processes with possibly defective reproduction laws. These results may serve as a stepping stone towards incisive general results for the Galton-Watson processes in a varying environment.

Galton-Watson theta-processes in a varying environment

Abstract

We consider a special class of Galton-Watson theta-processes in a varying environment fully defined by four parameters, with two of them being fixed over time , and the other two characterizing the altering reproduction laws. We establish a sequence of transparent limit theorems for the theta-processes with possibly defective reproduction laws. These results may serve as a stepping stone towards incisive general results for the Galton-Watson processes in a varying environment.
Paper Structure (6 sections, 18 theorems, 105 equations)

This paper contains 6 sections, 18 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Proper and defective reproduction laws
  3. Limit theorems for the proper GW$^{\,\theta}$-processes
  4. Limit theorems for the defective GW$^{\,\theta}$-process
  5. Examples
  6. Proofs

Key Result

Lemma 1

Consider a GW$^{\,\theta}$-process with parameters $(\theta,r,a_n,c_n)$. If $\theta\ne0$, then and if $\theta=0$, then Here,

Theorems & Definitions (22)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 12 more